Let f(x)=x^2-2x+1. Find all points on the graph of f(x) whose tangent line also passes through (3,1)

Question

Let f(x) = x^2 -2x+1. Find all points on the graph of f(x) whose
tangent line also passes through the point (3,1).

Answer 1

Point slope form is y-y1 = m(x-x1)

Add y1 to each side.

y = m(x-x1)+y1 ---> (1)

Where m is slope, (x1,y1) = (3,1) are point on the line.

Now find f'(x)

f(3) = 9-6+1 = 4

f(1) = 1-2+1 = 0

f'(x) = f(b)-f(a)/b-a

= (0-4)/(1-3)

= -4/-2

= 2

For a line tangent to a curve

Slope m = f'(x), (x1,y1) = (3,1)

From (1)

y = 2(x-3)+1

y = 2x-6+1

y = 2x-5

Graph